Descent Procedure
Here we will show that the intertwining operator of Eq. (18) can be interpreted as
the generalization of the string theory descent procedure, which relates unintegrated and integrated vertex operators.
1 Interpretation as a descent procedure
Consider an unintegrated vertex opearator
.
We interpret it as an element of the cohomology of
with some ghost number
:
Eq. (
18) allows us to construct from
a cohomology class of
,
where
is the Lie algebra cohomology differential of our Lie algebra
with coefficients
in
, as follows:
This expression is “inhomogeneous”, in the sense that different components have different ghost
numbers. Each application of
decreases the ghost number by one, but at the
same time rises the degree of the Lie algebra cochain. In the context of closed string,
the top component coincides with
, then goes
, then
and so on. In particular,
is our interpretation of the
integrated vertex operator.
We could have used
instead of
. We prefer
to use
because it leads to the
local result.
In string theory the use of such an inhomogeneous expression is often referred to as the “descent procedure”.
2 Integrated vertex and Lie algebra cohomology
We have shown that the cohomology of
is the same as the cohomology of
.
The cohomology of
can be computed using the spectral sequence, corresponding
to the filtration by the ghost number. Let
consist of the functions with
the ghost number
. At the first page, we have:
Therefore, if
, then the cohomology of
is equivalent to the
cohomology of
with values in
.
3 Comparison of and 
We have two operators satisfying the identical intertwining relations:
This suggests the existence of some operator
such that:
This
is an inhomogeneous operator-form:
TODO: write the full formula.
4 Relation between integrated and unintegrated vertices
Consider the special case of
flat worldsheet. There is a subalgebra
consisting of translations (
and
).
Let us restrict
to this subalgebra. This simplifies the computation because:
. This is true even at the level of cochains. Therefore we have:
Going back from the Faddeev-Popov notations to the form notations:
we obtain:
(here
and
is what remains of
).
This is the usual
integrated vertex operator of the bosonic string theory.
5 Various types of Lie algebra cohomology
We can consider three different cohomologies:
Cohomology of the Lie algebra of diffeomorphisms with coefficients in
Cohomology of the Virasoro algebra with zero central charge with coefficients in functions on a Lagrangian submanifold.
If we restrict the cocycle of the Lie algebra of diffeomorphisms to the subalgebra preserving the Lagrangian submanifold,
we get a cocycle of the Virasoro algebra. Again, we should go on-shell and take into account 
.
Cohomology of 
which computes the integrated vertex operators
TODO: work out the precise relation between these objects.
in closed string theory and
in open string theory
contains integration, the odd Poisson bracket is local (i.e. involves a delta-function) and therefore removes the integral.
